Высшая математика — Часть 5. Р.М. Жевняк, А.А. Карпук 1988

This book is the fifth and final part of the textbook Higher Mathematics for students of higher technical educational institutions. Like the previous four parts, it is based on lectures delivered by the authors to students of the Minsk Radio Engineering Institute. The book presents the fundamentals of the theory of difference equations, the Z-transform and the discrete Fourier transform, generalized functions, probability theory, mathematical statistics, and the foundations of the theory of stochastic processes, as well as elementary numerical methods.

Although the sections “Difference Equations,” “Z-Transform,” “Discrete Fourier Transform,” and “Generalized Functions” are not part of the standard curriculum, the authors considered it necessary to include them in this textbook because they are widely used by engineers and researchers in solving various applied problems. In particular, difference equations are used in the numerical solution of differential equations by finite-difference methods, while generalized functions are applied in probability theory, mathematical statistics, the theory of impulse systems, and other fields.

In view of the growing importance of probability theory, mathematical statistics, and the foundations of stochastic process theory in various branches of science and technology, these sections are presented in considerable detail. Furthermore, the curriculum in higher mathematics for engineering and technical specializations requires the completion of a standard computational assignment in probability theory and mathematical statistics. Therefore, the corresponding chapters of the textbook are written in such a way that a student who has mastered them will be fully capable of completing this assignment.

The principal numerical methods of higher mathematics are presented in strict accordance with the curriculum. The book contains numerous examples illustrating the main theoretical concepts. At the end of each section, problems and exercises are provided for independent study.

The numbering of chapters, sections, formulas, and figures continues the numbering used in the previous volumes. When referring to a formula or theorem, the number of the corresponding part of the textbook is indicated. The beginning of theorem proofs and the solutions of examples and problems are marked by the symbol □, while the end of a proof is marked by the symbol ■.

The authors express their gratitude to the reviewers of the book: Doctor of Physical and Mathematical Sciences, Professor M. V. Fedoryuk of the Department of Higher Mathematics at the Moscow Institute of Physics and Technology; Corresponding Member of the Academy of Sciences of the USSR, Head of the Department of Higher Mathematics at the Moscow Power Engineering Institute, Doctor of Physical and Mathematical Sciences, Professor S. I. Pokhozhaev; and Professor M. L. Krasnov of the same department, whose careful and objective review contributed greatly to improving the content of the book.

The authors hope that this final volume of the textbook will assist students in studying the specialized sections of higher mathematics taught in technical institutes. In their opinion, all five parts of the textbook together constitute a complete course in higher mathematics for students of technical higher educational institutions, designed for 510 instructional hours.

There is all 5 Parts combined volume published in 1988.

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 3

17. Difference Equations. Z-Transform. Generalized Functions . . . . . . . 5

17.1. Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

17.2. Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

17.3. Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

17.4. Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

18. Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18.1. Elements of Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18.2. Probability of an Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

18.3. Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

18.4. Sequence of Independent Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

18.5. Scalar Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

18.6. Vector Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

18.7. Numerical Characteristics of Scalar Random Variables . . . . . . . . . . . . . . . . 88

18.8. Numerical Characteristics of Vector Random Variables . . . . . . . . . . . . . . . 103

18.9. Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

18.10. Limit Theorems of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 114

18.11. Two-Dimensional Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 119

19. Elements of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . 125

19.1. Sample and its Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

19.2. Statistical Estimation of Distribution Parameters . . . . . . . . . . . . . . . . . . . 133

19.3. Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

19.4. Linear Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

20. Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

20.1. Finite Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

20.2. Basic Concepts of the Theory of Random Processes . . . . . . . . . . . . . . . . 166

20.3. Basic Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

21. Foundations of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 182

21.1. Approximate Numbers and Operations on Them . . . . . . . . . . . . . . . . . . . 182

21.2. Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

21.3. Interpolation of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

21.4. Solving Linear Systems by the Gauss Method . . . . . . . . . . . . . . . . . . . . 201

21.5. Iterative Methods for Solving Equations and Systems of Equations . . . . . . . 206

21.6. Approximate Calculation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 218

21.7. Numerical Solution of Ordinary Differential Equations . . . . . . . . . . . . . . . 225

21.8. Finite Difference Method for Solving Problems of Mathematical Physics . . . . 229

Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251